Gale Comparability Classifies the Generalized Dodgson Inequality for Totally Nonnegative Matrices

We give a complete classification of the generalized Dodgson inequality for minors of totally nonnegative matrices introduced by Danilov, Karzanov, and Koshevoy. For distinct equal-cardinality subsets A={a1<…<am} and B={b1<…<bm}, the generalized Dodgson inequality holds for every totally nonnegative matrix if and only if A and B are comparable in Gale (componentwise) order: either ar<=br for every r, or br<=ar for every r. The equal-set case A=B is not universal. The proof uses the published Danilov-Karzanov-Koshevoy matching-multiplicity theorem as its sole major external input. After that reduction, universality is characterized by a finite noncrossing-matching argument: a cross-half chord exists exactly when the prefix-difference walk |A∩[t]|-|B∩[t]| takes both positive and negative values. All new combinatorial arguments and the complete downstream classification have additionally been machine-checked in Lean 4, conditional only on the explicit published DKK theorem interface. The companion Lean formalization is archived as a separate Zenodo software record linked to this preprint. Within a documented public-literature search through 9 September 2026, no equivalent complete classification was located. This is a bounded novelty statement and not a claim of absolute historical priority.